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Physica Scripta

Operational approach to phase operators based on classical optics

J. W. Noh, A Fougères, L. Mandeļ

Published Jan 1, 1993• 22 citations
Publisher

Abstract

Our operational approach to the problem of identifying dynamical variables to represent the phase, or the sine and cosine of the phase, of a quantum field is reviewed. We start from classical optics and examine what is usually measured to determine the phase difference between two fields. We point out that difficulties in phase measurements exist even in the classical domain, and these difficulties persist when the classical relations are used as guides in the construction of phase operators. We find that different measurement procedures lead naturally to different phase operators, so that there appears to be no one "correct" phase operator. We carry out a series of experiments to measure the phase difference between two fields in various quantum states, and then compare the results with the predictions of our formalism and some other formalisms. We obtain good agreement between experiment and our theoretical approach in every case, so far without exception.

DOI
10.1088/0031-8949/1993/t48/004

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The correct description of the phase variable in quantum mechanics is a question rooted in its earliest formulations. The atom mechanics [1] of Bohr and Sommerfeld—the precursor of modern quantum mechanics—ascribes a central role to action-angle variables. However, Heisenberg's matrix mechanics and Schrödinger's wave mechanics are formulated [2] in terms of the canonical variables representing cartesian coordinates and momenta. Stimulated by this work London [3] attempted a reformulation of these theories in the previously favored action-angle variables. However, this attempt failed due to the difficulty in ascribing quantum operators to the angle variables of classical theory. Despite these difficulties London found [4] an operator representation of the complex exponential of these angle variables. The quantum phase of light made its first appearance in Dirac's classic paper [5] on the quantization of the radiation field. In contrast to modern methods, he constructed the annihilation and creation operators for each field mode from the corresponding amplitude and phase operators. Phase and its quantum nature acquired new significance with the development of lasers in the early sixties: theoretical investigations highlighted significant problems with Dirac's original proposal for the phase operator. A particularly elegant illustration of the difficulty was given by Louisell [6] in 1963. The advent of phase sensitive quantum noise as demonstrated experimentally in the production and detection of squeezed light [7] has created a new wave of interest in the nature of quantum optical phase leading to the discovery of the hermitian optical phase operator [8]. There has followed an explosion of theoretical activity in this area stimulating fresh experimental investigations. In preparing this special issue we have attempted to present the current state of this active and rapidly moving field. We have arranged the papers into what we hope is a coherent representation of the field. The first papers give a historical perspective and overview of current thinking. The two recent experimental investigations which follow are intimately connected to the phase space description of quantum mechanics based on quasi-probability distributions. The representa tion of phase via phase space and its connection with phase-dependent measurements and the phase operator are addressed in the next section. Some more formal considerations pertinent to phase are presented in the following section. Gravitational wave detection and optical communication have motivated the study of the limits of phase noise. Some recent investigations on such optimal phase states are presented. The issue concludes with two papers discussing the significance of phase in light-matter interactions. In concluding we express our gratitude to the authors of the papers in this volume for their efforts in preparing their high quality presentations.

Physica Scripta•Jan 1, 1993•74 citations
Our operational approach to the problem of identifying dynamical variables to represent the phase, or the sine and cosine of the phase, of a quantum field is reviewed. We start from classical optics and examine what is usually measured to determine the phase difference between two fields. We point out that difficulties in phase measurements exist even in the classical domain, and these difficulties persist when the classical relations are used as guides in the construction of phase operators. We find that different measurement procedures lead naturally to different phase operators, so that there appears to be no one "correct" phase operator. We carry out a series of experiments to measure the phase difference between two fields in various quantum states, and then compare the results with the predictions of our formalism and some other formalisms. We obtain good agreement between experiment and our theoretical approach in every case, so far without exception.
The problem of identifying the dynamic variable representing the phase difference between quantized electromagnetic fields is approached via the measurement process. By comparison with classical optics, certain operators corresponding to the measured sine and cosine of the phase difference are identified, but different measurement schemes lead naturally to different operators. First and second moments of the sine and cosine operators are calculated for a two-mode field in a coherent state. The theory is then tested in a photon counting experiment, for mean detected photon numbers varying from 30 to 10-2 which extends the range of previous measurements 300- fold. Good agreement is found between theory and experiment.
We examine the problem of determining the phase difference between two optical fields, first for classical and later for quantum fields, by reference to two simple measurement schemes that yield the sine and/or cosine of the phase difference between classical fields. We show that certain difficulties exist even within the framework of semiclassical radiation theory when the field is very weak, and particularly when amplitude and phase fluctuations are correlated. We find that a clear distinction has to be made between the measured values of the sine or cosine and the values that can be inferred from a series of repeated measurements. A corresponding distinction can be made also for a quantum field, although the interpretation is not the same. The dynamical variables chosen to represent the cosine and sine that emerge from the discussion of the measurement schemes commute when the sine and cosine are obtained together, but not when the measurement yields one or the other. These sine and cosine operators have well-defined values only when there is a large dispersion of the photon number. We arrive at expressions for the moments of the measured and of the inferred sines and cosines that differ from most previous treatments. The expressions are applied to optical fields in several different quantum states. Only for the Fock state and for the so-called phase state, which was treated recently at some length by Pegg and Barnett [Phys. Rev. A 39, 1665 (1989)], do the measured and the inferred moments coincide. Our analysis of the problem of phase measurement leads to the conclusion that the appropriate dynamical variables for the measured sine and cosine depend on the measurement scheme, and that different schemes correspond to different operators.
We address the problem of identifying the dynamical variables representing the phase of a quantum field by analyzing what is measured in two simple homodyne experiments. This leads us to identify operators C${\mathrm{^}}_{\mathit{M}}$,S${\mathrm{^}}_{\mathit{M}}$ corresponding to the measured cosine and sine of the phase difference. The predictions of the theory are tested in an experiment in which the input fields are derived from a highly stable He:Ne laser. Photon-counting measurements extending from mean photon numbers 30 down to ${10}^{\mathrm{\ensuremath{-}}2}$ show good agreement between theory and experiment. The implications for identifying the ``correct'' phase operator are briefly discussed.
We have previously developed a formalism for identifying quantum operators corresponding to the sine and cosine of the phase differences between two quantum fields.1 Our approach is based on what is actually measured in an experiment, and it leads to different operators for different measurement schemes. Predictions based on our theory differ from those based on the Susskind-Glogower and the Pegg-Barnett operators. We now present the results of new experiments designed to test the validity of our approach, including measurements of the higher moments of the sine and cosine for different quantum states. The results of all measurements confirm our formalism in every case.
The correct description of the phase variable in quantum mechanics is a question rooted in its earliest formulations. The atom mechanics [1] of Bohr and Sommerfeld—the precursor of modern quantum mechanics—ascribes a central role to action-angle variables. However, Heisenberg's matrix mechanics and Schrödinger's wave mechanics are formulated [2] in terms of the canonical variables representing cartesian coordinates and momenta. Stimulated by this work London [3] attempted a reformulation of these theories in the previously favored action-angle variables. However, this attempt failed due to the difficulty in ascribing quantum operators to the angle variables of classical theory. Despite these difficulties London found [4] an operator representation of the complex exponential of these angle variables. The quantum phase of light made its first appearance in Dirac's classic paper [5] on the quantization of the radiation field. In contrast to modern methods, he constructed the annihilation and creation operators for each field mode from the corresponding amplitude and phase operators. Phase and its quantum nature acquired new significance with the development of lasers in the early sixties: theoretical investigations highlighted significant problems with Dirac's original proposal for the phase operator. A particularly elegant illustration of the difficulty was given by Louisell [6] in 1963. The advent of phase sensitive quantum noise as demonstrated experimentally in the production and detection of squeezed light [7] has created a new wave of interest in the nature of quantum optical phase leading to the discovery of the hermitian optical phase operator [8]. There has followed an explosion of theoretical activity in this area stimulating fresh experimental investigations. In preparing this special issue we have attempted to present the current state of this active and rapidly moving field. We have arranged the papers into what we hope is a coherent representation of the field. The first papers give a historical perspective and overview of current thinking. The two recent experimental investigations which follow are intimately connected to the phase space description of quantum mechanics based on quasi-probability distributions. The representa tion of phase via phase space and its connection with phase-dependent measurements and the phase operator are addressed in the next section. Some more formal considerations pertinent to phase are presented in the following section. Gravitational wave detection and optical communication have motivated the study of the limits of phase noise. Some recent investigations on such optimal phase states are presented. The issue concludes with two papers discussing the significance of phase in light-matter interactions. In concluding we express our gratitude to the authors of the papers in this volume for their efforts in preparing their high quality presentations.